On the torsion of functionally graded anisotropic linearly elastic bars

The torsion of homogeneous, isotropic, linearly elastic cylindrical bars has been the subject of numerous investigations from theoretical, computational and applied viewpoints. It is well known that in this case, the circular shaft is the only one whose cross-section does not involve a warping displacement in the axial direction. For an inhomogeneous, isotropic, circular bar, there is also no warping of the cross-section provided that the shear modulus varies only in the radial direction. For general anisotropic materials, torsion induces bending and vice versa. For those anisotropic materials with at least one plane of elastic symmetry normal to the axial direction, pure torsion is possible. For homogeneous materials of this type and special inhomogeneous materials, it has been shown by various arguments that no warping occurs for bars of particular elliptical cross-section. Here, we provide a unified derivation of the foregoing results. Some discussion of torsional rigidities is also given.